Perform the following arithmetic operations. Reduce your final answer to lowest terms. Express a negative fraction by including "-" in the numerator. \( -\left(\frac{2}{3} \times 2 \frac{1}{2}\right)-\left(1 \frac{1}{8} \div-6\right)+\left(-1 \frac{5}{6}\right)\left(-\frac{9}{11}\right) \)

\[ -\left(\frac{2}{3} \times 2\,\frac{1}{2}\right) - \left(1\,\frac{1}{8} \div -6 \right) + \left(-1\,\frac{5}{6}\right)\left(-\frac{9}{11}\right) \] 1. Convert mixed numbers to improper fractions: \[ 2\,\frac{1}{2} = \frac{5}{2}, \quad 1\,\frac{1}{8} = \frac{9}{8}, \quad -1\,\frac{5}{6} = -\frac{11}{6} \] 2. Substitute into the expression: \[ -\left(\frac{2}{3} \times \frac{5}{2}\right) - \left(\frac{9}{8} \div -6\right) + \left(-\frac{11}{6}\right)\left(-\frac{9}{11}\right) \] 3. Simplify each term separately: a. First term: \[ \frac{2}{3} \times \frac{5}{2} = \frac{2 \times 5}{3 \times 2} = \frac{10}{6} = \frac{5}{3} \] Applying the negative sign: \[ -\frac{5}{3} \] b. Second term: \[ \frac{9}{8} \div -6 = \frac{9}{8} \times \left(-\frac{1}{6}\right) = -\frac{9}{48} = -\frac{3}{16} \] Then, with the subtraction outside: \[ -\left(-\frac{3}{16}\right) = \frac{3}{16} \] c. Third term: \[ \left(-\frac{11}{6}\right)\left(-\frac{9}{11}\right) = \frac{11 \times 9}{6 \times 11} = \frac{9}{6} = \frac{3}{2} \] 4. Now, combine the simplified terms: \[ -\frac{5}{3} + \frac{3}{16} + \frac{3}{2} \] 5. Find a common denominator (which is 48): \[ -\frac{5}{3} = -\frac{5 \times 16}{48} = -\frac{80}{48} \] \[ \frac{3}{16} = \frac{3 \times 3}{48} = \frac{9}{48} \] \[ \frac{3}{2} = \frac{3 \times 24}{48} = \frac{72}{48} \] 6. Sum the fractions: \[ -\frac{80}{48} + \frac{9}{48} + \frac{72}{48} = \frac{-80 + 9 + 72}{48} = \frac{1}{48} \] The final answer in lowest terms is: \[ \frac{1}{48} \]